TITLE: Intersection of subvarieties versus ranks of Chow groups modulo numerical equivalences
QUESTION [13 upvotes]: A nice property of $\mathbb P^n$ is:

Property 1: Two subvarieties $U,V$ such that $\operatorname{dim} U +\operatorname{dim} V \geq n$ always intersect.

(for example, any 2 curves in $\mathbb P^2$ intersect)
There are other  smooth varieties $X$ when Properties 1 holds. For example, a sufficient condition is that the ranks of $\text{CH}^i_{num}(X)$ are $1$ for $i\leq n/2$. Here $n = \operatorname{dim} X$ and $\text{CH}^i_{num}(X)$ is the Chow group of codimension $i$ modulo numerical equivalences. 
My question is whether some converse is true:

Question: Let $X$ be a smooth projective variety satisfying  Property 1. Does that impose some upper bounds on the ranks of $\text{CH}^i_{num}(X)$  for $i\leq n/2$? 

Let's assume we are over $\mathbb C$, but I am also interested in results over any ground fields.
One can ask the same questions for the ranks of $\text{CH}^i_{hom}(X)$ (I think they are conjectured to be the same). The baby case is $i=1$, where the question asks if Property 1 tells us something about the rank of the Neron-Severi group of $X$. 
I am aware that the question is a little vague (upper bound as function of what?), but that was because of my ignorance, so comments to improve the question are welcome.

REPLY [7 votes]: This is a very interesting question and I guess that a general answer is unknown, already in the case $i=1$. Let me just make the following
Remark. There exists no upper bound on $\textrm{rank } NS(X)$ which is independent on the dimension.
In fact, let us consider a complex Abelian variety $X$ of dimension $g$ such that $End_{\mathbb{Q}}(X)$ is a totally real number field of degree $g$ over $\mathbb{Q}$. These 
varieties do exist and the general one is simple, see [Birkenhake - Lange, Chapters 5 and 9].
Therefore it is known that
$\rho(X) =\textrm{rank } \textrm{NS}(X) = \textrm{rank } End^s_{\mathbb{Q}}(X)=g$,
where $End^s_{\mathbb{Q}}(X)$ denotes the subgroups of elements in $End_{\mathbb{Q}}(X)$ which are symmetric with respect to the Rosati involution.
On the other hand, in a simple Abelian variety any effective divisor is ample, so two effective divisors always intersect and $X$  satisfies Property 1.